Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

TL;DR

This paper explores the connection between exactly solved edge-interaction models satisfying the star-triangle relation and classical integrable systems on planar graphs, revealing a unifying master solution and its low-temperature limit.

Contribution

It demonstrates that all known solutions of the star-triangle relation derive from a single elliptic gamma function-based master solution and links quantum models to classical integrable systems.

Findings

All solutions are special cases of a master elliptic gamma function solution.

Low-temperature limit reproduces classical integrable systems on graphs.

Invariance under star-triangle transformations relates to Z-invariance of lattice models.

Abstract

In this paper we give an overview of exactly solved edge-interaction models, where the spins are placed on sites of a planar lattice and interact through edges connecting the sites. We only consider the case of a single spin degree of freedom at each site of the lattice. The Yang-Baxter equation for such models takes a particular simple form called the star-triangle relation. Interestigly all known solutions of this relation can be obtained as particular cases of a single "master solution", which is expressed through the elliptic gamma function and have continuous spins taking values of the circle. We show that in the low-temperature (or quasi-classical) limit these lattice models reproduce classical discrete integrable systems on planar graphs previously obtained and classified by Adler, Bobenko and Suris through the consistency-around-a-cube approach. We also discuss inversion relations, the physicical meaning of Baxter's rapidity-independent parameter in the star-triangle relations and the invariance of the action of the classical systems under the star-triangle (or cube-flip) transformation of the lattice, which is a direct consequence of Baxter's Z-invariance in the associated lattice models.